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Abu Bakr al-Baghdadi last month released an audio message urging his supporters to defend the Iraqi city of Mosul

The Pentagon said on Friday it believes that Islamic State chief Abu Bakr al-Baghdadi is indeed alive, despite repeated efforts by the US-led coalition to take out the jihadist group leader.

Baghdadi has kept a low profile, despite having declared himself the leader of a renewed Muslim caliphate, but last month released a defiant audio message urging his supporters to defend the northern Iraqi city of Mosul.

It is not clear if he is in the besieged city, where he declared his caliphate in 2014 after the IS group seized territory covering much of eastern Syria and northern Iraq.

"We do think Baghdadi is alive and is still leading ISIL and we are obviously doing everything we can to track his movements," Pentagon spokesman Peter Cook told CNN.

"If we get the opportunity, we certainly would take advantage of any opportunity to deliver him the justice he deserves," he said.

"We're doing everything we can. This is something we're spending a lot of time on."

In mid-December, the United States more than doubled the bounty on the shadowy IS leader's head to $25 million.

The group has only released one video of Baghdadi, showing a man with a black and grey beard wearing a black robe and matching turban, dating back to 2014.

Cook suggested that Baghdadi is isolated because coalition raids have killed many IS leaders.

"He's having a hard time finding advisers and confidants to speak with because a lot of them are no longer with us," the spokesman said.

According to an official Iraqi government document, Baghdadi was born in Samarra in 1971. He apparently joined the insurgency that erupted after the 2003 US-led invasion of Iraq and spent time in an American military prison.